长长的同一节点值路径——二叉树中的最大单调路径

在计算机科学中，二叉树是一种数据结构，它将每个节点最多分为两个子树，左子树和右子树。我们把一个顶点作为根，从根节点到叶子的路径称为根叶路径。我们这里所说的单调路径是指，从根节点到叶子的路径上，每个节点的值都相同，这个路径称为单调路径。

题目要求：
给定一个二叉树，要求找出其中最长的单调路径，这个路径可以经过或不经过根节点。

方法：
我们可以用深度优先搜索的方法来解决这个问题，从每个节点开始向下探索，如果某个节点的值与父节点的值相同，那么我们可以继续向下探索，直到遇到不同的值或者遇到叶节点。在探索的过程中，我们记录下当前路径的长度，并与之前记录的最长路径长度进行比较，如果当前路径长度大于之前记录的最长路径长度，那么我们更新最长路径长度。

我们也可以使用递归的方法来解决这个问题，递归函数的参数是一个节点，递归函数首先检查当前节点的值是否与父节点的值相同，如果相同，那么递归函数继续向下探索，如果不同，那么递归函数返回当前路径的长度。在递归函数返回的时候，我们比较当前路径长度与之前记录的最长路径长度，如果当前路径长度大于之前记录的最长路径长度，那么我们更新最长路径长度。

代码示例：

def longest_monotonic_path(root):
    """
    Finds the length of the longest path in a binary tree where each node in the path has the same value.

    Args:
        root: The root node of the binary tree.

    Returns:
        The length of the longest path.
    """

    # Initialize the longest path length to 0.
    longest_path = 0

    # Call the recursive function to find the longest path.
    longest_path = _longest_monotonic_path(root, longest_path)

    # Return the longest path length.
    return longest_path


def _longest_monotonic_path(node, longest_path):
    """
    Finds the length of the longest path in a binary tree where each node in the path has the same value.

    Args:
        node: The current node in the binary tree.
        longest_path: The length of the longest path found so far.

    Returns:
        The length of the longest path.
    """

    # Check if the current node is None.
    if node is None:
        return 0

    # Check if the current node is a leaf node.
    if node.left is None and node.right is None:
        return 1

    # Check if the current node is the root node.
    if node is root:
        # Check if the left and right subtrees have the same value.
        if node.left is not None and node.left.val == node.val:
            # Calculate the length of the longest path in the left subtree.
            left_path = _longest_monotonic_path(node.left, longest_path)

            # Calculate the length of the longest path in the right subtree.
            right_path = _longest_monotonic_path(node.right, longest_path)

            # Update the longest path length.
            longest_path = max(longest_path, left_path + right_path + 1)

        # Check if the left and right subtrees have different values.
        if node.left is not None and node.left.val != node.val:
            # Calculate the length of the longest path in the left subtree.
            left_path = _longest_monotonic_path(node.left, longest_path)

            # Update the longest path length.
            longest_path = max(longest_path, left_path)

        # Check if the left and right subtrees have different values.
        if node.right is not None and node.right.val != node.val:
            # Calculate the length of the longest path in the right subtree.
            right_path = _longest_monotonic_path(node.right, longest_path)

            # Update the longest path length.
            longest_path = max(longest_path, right_path)

        # Return the longest path length.
        return longest_path

    # Check if the left and right subtrees have the same value.
    if node.left is not None and node.left.val == node.val:
        # Calculate the length of the longest path in the left subtree.
        left_path = _longest_monotonic_path(node.left, longest_path)

        # Update the longest path length.
        longest_path = max(longest_path, left_path + 1)

    # Check if the left and right subtrees have different values.
    if node.left is not None and node.left.val != node.val:
        # Calculate the length of the longest path in the left subtree.
        left_path = _longest_monotonic_path(node.left, longest_path)

        # Update the longest path length.
        longest_path = max(longest_path, left_path)

    # Check if the left and right subtrees have different values.
    if node.right is not None and node.right.val != node.val:
        # Calculate the length of the longest path in the right subtree.
        right_path = _longest_monotonic_path(node.right, longest_path)

        # Update the longest path length.
        longest_path = max(longest_path, right_path)

    # Return the longest path length.
    return longest_path