https://leetcode.com/problems/lowest-common-ancestor-of-a-binary-tree/#/description

Given a binary tree, find the lowest common ancestor (LCA) of two given nodes in the tree.

According to the definition of LCA on Wikipedia: “The lowest common ancestor is defined between two nodes v and w as the lowest node in T that has both v and w as descendants (where we allow a node to be a descendant of itself).”

        _______3______
       /                  ___5__          ___1__
   /      \        /         6      _2       0       8
         /           7   4

For example, the lowest common ancestor (LCA) of nodes 5 and 1 is 3. Another example is LCA of nodes 5and 4 is 5, since a node can be a descendant of itself according to the LCA definition.

Sol:

Bottom-up.

Find where is p or q. 

1) if p or q is the root, return root

2) if p and q are on both sides of the tree, return root

3) if p and q are on left subtree, return left node.

4) if p and q are on right subtree, return right node.

To make the logic of 3) and 4) more apprent to computers, we use left/right variables to check if p and q exist on left/right subtree. # simple discrete mathematics 

# Definition for a binary tree node.
# class TreeNode(object):
#     def __init__(self, x):
#         self.val = x
#         self.left = None
#         self.right = None

class Solution(object):
    def lowestCommonAncestor(self, root, p, q):
        """
        :type root: TreeNode
        :type p: TreeNode
        :type q: TreeNode
        :rtype: TreeNode
        """
        # if root is null or find p or q
        if root == None or root == q or root == p:
            return root
        # find p or q in the left subtree
        left = self.lowestCommonAncestor(root.left, p, q)
        # find p or q in the right subtree
        right = self.lowestCommonAncestor(root.right, p, q)
        
        if left and right:
            return root
        else:
            if not left:
                return right
            else:
                return left

Similar Problem：

235. Lowest Common Ancestor of a Binary Search Tree

In the BSTproblem, we use the one BST attribute, values of p and q, to check which side p and q are on. We can also use iteration on it.   

236. Lowest Common Ancestor of a Binary Tree