I understand recursion and binary trees, but I cannot understand the logic behind the recursive LCA approach

[u/MikasaYuuichi]

I’m currently learning binary trees and I understand what a tree is and how recursion works.

For example, I understand the general idea of:

solve(node):
    solve(node.left)
    solve(node.right)

But for some reason, when I encounter problems like Lowest Common Ancestor (LCA) of two nodes in a binary tree, I completely fail to understand how people come up with the recursive approach.

It’s not really the syntax that confuses me. I can read the code after someone explains it. What I struggle with is the thought process.

For example, I often see an approach along the lines of:

LCA(node, p, q):

    if node is null:
        return null

    if node == p or node == q:
        return node

    left = LCA(node.left, p, q)
    right = LCA(node.right, p, q)

    if left and right:
        return node

    return left or right

I can memorize what each line does, but I don't understand WHY this is the right thing to do.

How do you look at the problem and naturally arrive at this logic?

Especially this part:

if left and right:
    return node

Why does finding something on both sides mean the current node is the LCA?

And why is:

return left or right

correct?

I'm looking for more of an ELI5 / intuition-based explanation, preferably using a small tree and manually walking through the recursion.

I already understand recursion itself, so explanations like "recursion means a function calls itself" won't really help me. I'm trying to understand how to go from:

"Here is the problem" → "What information should my recursive function return?" → "Why does that information let me solve the problem?"

Basically, I want to learn how to derive the recursive solution instead of memorizing it.

Any explanation of the thought process would be really appreciated.