hello-algo/en/docs/chapter_tree/binary_tree.md

# Binary tree

A <u>binary tree</u> is a non-linear data structure that represents the hierarchical relationship between ancestors and descendants and embodies the divide-and-conquer logic of "splitting into two". Similar to a linked list, the basic unit of a binary tree is a node, and each node contains a value, a reference to its left child node, and a reference to its right child node.

=== "Python"

    ```python title=""
    class TreeNode:
        """Binary tree node"""
        def __init__(self, val: int):
            self.val: int = val                # Node value
            self.left: TreeNode | None = None  # Reference to left child node
            self.right: TreeNode | None = None # Reference to right child node
    ```

=== "C++"

    ```cpp title=""
    /* Binary tree node */
    struct TreeNode {
        int val;          // Node value
        TreeNode *left;   // Pointer to left child node
        TreeNode *right;  // Pointer to right child node
        TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
    };
    ```

=== "Java"

    ```java title=""
    /* Binary tree node */
    class TreeNode {
        int val;         // Node value
        TreeNode left;   // Reference to left child node
        TreeNode right;  // Reference to right child node
        TreeNode(int x) { val = x; }
    }
    ```

=== "C#"

    ```csharp title=""
    /* Binary tree node */
    class TreeNode(int? x) {
        public int? val = x;    // Node value
        public TreeNode? left;  // Reference to left child node
        public TreeNode? right; // Reference to right child node
    }
    ```

=== "Go"

    ```go title=""
    /* Binary tree node */
    type TreeNode struct {
        Val   int
        Left  *TreeNode
        Right *TreeNode
    }
    /* Constructor */
    func NewTreeNode(v int) *TreeNode {
        return &TreeNode{
            Left:  nil, // Pointer to left child node
            Right: nil, // Pointer to right child node
            Val:   v,   // Node value
        }
    }
    ```

=== "Swift"

    ```swift title=""
    /* Binary tree node */
    class TreeNode {
        var val: Int // Node value
        var left: TreeNode? // Reference to left child node
        var right: TreeNode? // Reference to right child node

        init(x: Int) {
            val = x
        }
    }
    ```

=== "JS"

    ```javascript title=""
    /* Binary tree node */
    class TreeNode {
        val; // Node value
        left; // Pointer to left child node
        right; // Pointer to right child node
        constructor(val, left, right) {
            this.val = val === undefined ? 0 : val;
            this.left = left === undefined ? null : left;
            this.right = right === undefined ? null : right;
        }
    }
    ```

=== "TS"

    ```typescript title=""
    /* Binary tree node */
    class TreeNode {
        val: number;
        left: TreeNode | null;
        right: TreeNode | null;
    
        constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {
            this.val = val === undefined ? 0 : val; // Node value
            this.left = left === undefined ? null : left; // Reference to left child node
            this.right = right === undefined ? null : right; // Reference to right child node
        }
    }
    ```

=== "Dart"

    ```dart title=""
    /* Binary tree node */
    class TreeNode {
      int val;         // Node value
      TreeNode? left;  // Reference to left child node
      TreeNode? right; // Reference to right child node
      TreeNode(this.val, [this.left, this.right]);
    }
    ```

=== "Rust"

    ```rust title=""
    use std::rc::Rc;
    use std::cell::RefCell;

    /* Binary tree node */
    struct TreeNode {
        val: i32,                               // Node value
        left: Option<Rc<RefCell<TreeNode>>>,    // Reference to left child node
        right: Option<Rc<RefCell<TreeNode>>>,   // Reference to right child node
    }

    impl TreeNode {
        /* Constructor */
        fn new(val: i32) -> Rc<RefCell<Self>> {
            Rc::new(RefCell::new(Self {
                val,
                left: None,
                right: None
            }))
        }
    }
    ```

=== "C"

    ```c title=""
    /* Binary tree node */
    typedef struct TreeNode {
        int val;                // Node value
        int height;             // Node height
        struct TreeNode *left;  // Pointer to left child node
        struct TreeNode *right; // Pointer to right child node
    } TreeNode;

    /* Constructor */
    TreeNode *newTreeNode(int val) {
        TreeNode *node;

        node = (TreeNode *)malloc(sizeof(TreeNode));
        node->val = val;
        node->height = 0;
        node->left = NULL;
        node->right = NULL;
        return node;
    }
    ```

=== "Kotlin"

    ```kotlin title=""
    /* Binary tree node */
    class TreeNode(val _val: Int) {  // Node value
        val left: TreeNode? = null   // Reference to left child node
        val right: TreeNode? = null  // Reference to right child node
    }
    ```

=== "Ruby"

    ```ruby title=""

    ```

=== "Zig"

    ```zig title=""

    ```

Each node has two references (pointers), pointing respectively to the <u>left-child node</u> and <u>right-child node</u>. This node is called the <u>parent node</u> of these two child nodes. When given a node of a binary tree, we call the tree formed by this node's left child and all nodes below it the <u>left subtree</u> of this node. Similarly, the <u>right subtree</u> can be defined.

**In a binary tree, except leaf nodes, all other nodes contain child nodes and non-empty subtrees.** As shown in the figure below, if "Node 2" is regarded as a parent node, its left and right child nodes are "Node 4" and "Node 5" respectively. The left subtree is formed by "Node 4" and all nodes beneath it, while the right subtree is formed by "Node 5" and all nodes beneath it.

![Parent Node, child Node, subtree](binary_tree.assets/binary_tree_definition.png)

## Common terminology of binary trees

The commonly used terminology of binary trees is shown in the figure below.

- <u>Root node</u>: The node at the top level of a binary tree, which does not have a parent node.
- <u>Leaf node</u>: A node that does not have any child nodes, with both of its pointers pointing to `None`.
- <u>Edge</u>: A line segment that connects two nodes, representing a reference (pointer) between the nodes.
- The <u>level</u> of a node: It increases from top to bottom, with the root node being at level 1.
- The <u>degree</u> of a node: The number of child nodes that a node has. In a binary tree, the degree can be 0, 1, or 2.
- The <u>height</u> of a binary tree: The number of edges from the root node to the farthest leaf node.
- The <u>depth</u> of a node: The number of edges from the root node to the node.
- The <u>height</u> of a node: The number of edges from the farthest leaf node to the node.

![Common Terminology of Binary Trees](binary_tree.assets/binary_tree_terminology.png)

!!! tip

    Please note that we usually define "height" and "depth" as "the number of edges traversed", but some questions or textbooks may define them as "the number of nodes traversed". In this case, both height and depth need to be incremented by 1.

## Basic operations of binary trees

### Initializing a binary tree

Similar to a linked list, the initialization of a binary tree involves first creating the nodes and then establishing the references (pointers) between them.

=== "Python"

    ```python title="binary_tree.py"
    # Initializing a binary tree
    # Initializing nodes
    n1 = TreeNode(val=1)
    n2 = TreeNode(val=2)
    n3 = TreeNode(val=3)
    n4 = TreeNode(val=4)
    n5 = TreeNode(val=5)
    # Linking references (pointers) between nodes
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5
    ```

=== "C++"

    ```cpp title="binary_tree.cpp"
    /* Initializing a binary tree */
    // Initializing nodes
    TreeNode* n1 = new TreeNode(1);
    TreeNode* n2 = new TreeNode(2);
    TreeNode* n3 = new TreeNode(3);
    TreeNode* n4 = new TreeNode(4);
    TreeNode* n5 = new TreeNode(5);
    // Linking references (pointers) between nodes
    n1->left = n2;
    n1->right = n3;
    n2->left = n4;
    n2->right = n5;
    ```

=== "Java"

    ```java title="binary_tree.java"
    // Initializing nodes
    TreeNode n1 = new TreeNode(1);
    TreeNode n2 = new TreeNode(2);
    TreeNode n3 = new TreeNode(3);
    TreeNode n4 = new TreeNode(4);
    TreeNode n5 = new TreeNode(5);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "C#"

    ```csharp title="binary_tree.cs"
    /* Initializing a binary tree */
    // Initializing nodes
    TreeNode n1 = new(1);
    TreeNode n2 = new(2);
    TreeNode n3 = new(3);
    TreeNode n4 = new(4);
    TreeNode n5 = new(5);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "Go"

    ```go title="binary_tree.go"
    /* Initializing a binary tree */
    // Initializing nodes
    n1 := NewTreeNode(1)
    n2 := NewTreeNode(2)
    n3 := NewTreeNode(3)
    n4 := NewTreeNode(4)
    n5 := NewTreeNode(5)
    // Linking references (pointers) between nodes
    n1.Left = n2
    n1.Right = n3
    n2.Left = n4
    n2.Right = n5
    ```

=== "Swift"

    ```swift title="binary_tree.swift"
    // Initializing nodes
    let n1 = TreeNode(x: 1)
    let n2 = TreeNode(x: 2)
    let n3 = TreeNode(x: 3)
    let n4 = TreeNode(x: 4)
    let n5 = TreeNode(x: 5)
    // Linking references (pointers) between nodes
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5
    ```

=== "JS"

    ```javascript title="binary_tree.js"
    /* Initializing a binary tree */
    // Initializing nodes
    let n1 = new TreeNode(1),
        n2 = new TreeNode(2),
        n3 = new TreeNode(3),
        n4 = new TreeNode(4),
        n5 = new TreeNode(5);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "TS"

    ```typescript title="binary_tree.ts"
    /* Initializing a binary tree */
    // Initializing nodes
    let n1 = new TreeNode(1),
        n2 = new TreeNode(2),
        n3 = new TreeNode(3),
        n4 = new TreeNode(4),
        n5 = new TreeNode(5);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "Dart"

    ```dart title="binary_tree.dart"
    /* Initializing a binary tree */
    // Initializing nodes
    TreeNode n1 = new TreeNode(1);
    TreeNode n2 = new TreeNode(2);
    TreeNode n3 = new TreeNode(3);
    TreeNode n4 = new TreeNode(4);
    TreeNode n5 = new TreeNode(5);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "Rust"

    ```rust title="binary_tree.rs"
    // Initializing nodes
    let n1 = TreeNode::new(1);
    let n2 = TreeNode::new(2);
    let n3 = TreeNode::new(3);
    let n4 = TreeNode::new(4);
    let n5 = TreeNode::new(5);
    // Linking references (pointers) between nodes
    n1.borrow_mut().left = Some(n2.clone());
    n1.borrow_mut().right = Some(n3);
    n2.borrow_mut().left = Some(n4);
    n2.borrow_mut().right = Some(n5);
    ```

=== "C"

    ```c title="binary_tree.c"
    /* Initializing a binary tree */
    // Initializing nodes
    TreeNode *n1 = newTreeNode(1);
    TreeNode *n2 = newTreeNode(2);
    TreeNode *n3 = newTreeNode(3);
    TreeNode *n4 = newTreeNode(4);
    TreeNode *n5 = newTreeNode(5);
    // Linking references (pointers) between nodes
    n1->left = n2;
    n1->right = n3;
    n2->left = n4;
    n2->right = n5;
    ```

=== "Kotlin"

    ```kotlin title="binary_tree.kt"
    // Initializing nodes
    val n1 = TreeNode(1)
    val n2 = TreeNode(2)
    val n3 = TreeNode(3)
    val n4 = TreeNode(4)
    val n5 = TreeNode(5)
    // Linking references (pointers) between nodes
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5
    ```

=== "Ruby"

    ```ruby title="binary_tree.rb"

    ```

=== "Zig"

    ```zig title="binary_tree.zig"

    ```

??? pythontutor "Code visualization"

    https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5&cumulative=false&curInstr=3&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false

### Inserting and removing nodes

Similar to a linked list, inserting and removing nodes in a binary tree can be achieved by modifying pointers. The figure below provides an example.

![Inserting and removing nodes in a binary tree](binary_tree.assets/binary_tree_add_remove.png)

=== "Python"

    ```python title="binary_tree.py"
    # Inserting and removing nodes
    p = TreeNode(0)
    # Inserting node P between n1 -> n2
    n1.left = p
    p.left = n2
    # Removing node P
    n1.left = n2
    ```

=== "C++"

    ```cpp title="binary_tree.cpp"
    /* Inserting and removing nodes */
    TreeNode* P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1->left = P;
    P->left = n2;
    // Removing node P
    n1->left = n2;
    ```

=== "Java"

    ```java title="binary_tree.java"
    TreeNode P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;
    ```

=== "C#"

    ```csharp title="binary_tree.cs"
    /* Inserting and removing nodes */
    TreeNode P = new(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;
    ```

=== "Go"

    ```go title="binary_tree.go"
    /* Inserting and removing nodes */
    // Inserting node P between n1 and n2
    p := NewTreeNode(0)
    n1.Left = p
    p.Left = n2
    // Removing node P
    n1.Left = n2
    ```

=== "Swift"

    ```swift title="binary_tree.swift"
    let P = TreeNode(x: 0)
    // Inserting node P between n1 and n2
    n1.left = P
    P.left = n2
    // Removing node P
    n1.left = n2
    ```

=== "JS"

    ```javascript title="binary_tree.js"
    /* Inserting and removing nodes */
    let P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;
    ```

=== "TS"

    ```typescript title="binary_tree.ts"
    /* Inserting and removing nodes */
    const P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;
    ```

=== "Dart"

    ```dart title="binary_tree.dart"
    /* Inserting and removing nodes */
    TreeNode P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;
    ```

=== "Rust"

    ```rust title="binary_tree.rs"
    let p = TreeNode::new(0);
    // Inserting node P between n1 and n2
    n1.borrow_mut().left = Some(p.clone());
    p.borrow_mut().left = Some(n2.clone());
    // Removing node P
    n1.borrow_mut().left = Some(n2);
    ```

=== "C"

    ```c title="binary_tree.c"
    /* Inserting and removing nodes */
    TreeNode *P = newTreeNode(0);
    // Inserting node P between n1 and n2
    n1->left = P;
    P->left = n2;
    // Removing node P
    n1->left = n2;
    ```

=== "Kotlin"

    ```kotlin title="binary_tree.kt"
    val P = TreeNode(0)
    // Inserting node P between n1 and n2
    n1.left = P
    P.left = n2
    // Removing node P
    n1.left = n2
    ```

=== "Ruby"

    ```ruby title="binary_tree.rb"

    ```

=== "Zig"

    ```zig title="binary_tree.zig"

    ```

??? pythontutor "Code visualization"

    https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E4%B8%8E%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20n2&cumulative=false&curInstr=37&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false

!!! tip

    It should be noted that inserting nodes may change the original logical structure of the binary tree, while removing nodes typically involves removing the node and all its subtrees. Therefore, in a binary tree, insertion and removal are usually performed through a set of operations to achieve meaningful outcomes.

## Common types of binary trees

### Perfect binary tree

As shown in the figure below, in a <u>perfect binary tree</u>, all levels are completely filled with nodes. In a perfect binary tree, leaf nodes have a degree of $0$, while all other nodes have a degree of $2$. The total number of nodes can be calculated as $2^{h+1} - 1$, where $h$ is the height of the tree. This exhibits a standard exponential relationship, reflecting the common phenomenon of cell division in nature.

!!! tip

    Please note that in the Chinese community, a perfect binary tree is often referred to as a <u>full binary tree</u>.

![Perfect binary tree](binary_tree.assets/perfect_binary_tree.png)

### Complete binary tree

As shown in the figure below, a <u>complete binary tree</u> is a binary tree where only the nodes in the bottom level are not completely filled, and the nodes in the bottom level are filled from left to right as much as possible. Please note that a perfect binary tree is also a complete binary tree.

![Complete binary tree](binary_tree.assets/complete_binary_tree.png)

### Full binary tree

As shown in the figure below, a <u>full binary tree</u>, except for the leaf nodes, has two child nodes for all other nodes.

![Full binary tree](binary_tree.assets/full_binary_tree.png)

### Balanced binary tree

As shown in the figure below, in a <u>balanced binary tree</u>, the absolute difference between the height of the left and right subtrees of any node does not exceed 1.

![Balanced binary tree](binary_tree.assets/balanced_binary_tree.png)

## Degeneration of binary trees

The figure below shows the ideal and degenerate structures of binary trees. A binary tree becomes a "perfect binary tree" when every level is filled; while it degenerates into a "linked list" when all nodes are biased toward one side.

- A perfect binary tree is an ideal scenario where the "divide and conquer" advantage of a binary tree can be fully utilized.
- On the other hand, a linked list represents another extreme where all operations become linear, resulting in a time complexity of $O(n)$.

![The Best and Worst Structures of Binary Trees](binary_tree.assets/binary_tree_best_worst_cases.png)

As shown in the table below, in the best and worst structures, the binary tree achieves either maximum or minimum values for leaf node count, total number of nodes, and height.

<p align="center"> Table <id> &nbsp; The Best and Worst Structures of Binary Trees </p>

|                                                 | Perfect binary tree | Linked list |
| ----------------------------------------------- | ------------------- | ----------- |
| Number of nodes at level $i$                    | $2^{i-1}$           | $1$         |
| Number of leaf nodes in a tree with height $h$  | $2^h$               | $1$         |
| Total number of nodes in a tree with height $h$ | $2^{h+1} - 1$       | $h + 1$     |
| Height of a tree with $n$ total nodes           | $\log_2 (n+1) - 1$  | $n - 1$     |