You are given the root of a binary search tree (BST) and an integer val.

Find the node in the BST that the node's value equals val and return the subtree rooted with that node. If such a node does not exist, return null.

I have came up with a recursive solution in python, but I also want to document down the mathematical representation. I came up with the following, am I correct notation wise?

$$ f(v, x) = \begin{cases} \emptyset & \text{if } v = \emptyset \\ f\Big(\ell(v), x\Big) & \text{if } g(v) > x \\ f\Big(r(v), x\Big) & \text{if } g(v) < x \\ v & \text{otherwise} \end{cases} $$


  • $\emptyset$ is the empty tree, or None in Python
  • $v$ is a node in the tree
  • $\ell(v)$ is the left child of $v$
  • $r(v)$ is the right child of $v$
  • $g(v)$ is the value of node $v$, and $x$ is the target value to search for.

In this formulation, the function $f$ takes two arguments: a node $v$ and a target value $x$. The function $\ell(v)$ returns the left child of the node $v$ and $r(v)$ returns the right child. Depending on the comparison of the value of node $v$ and the target $x$, the function $f$ recursively calls itself on either the left child or the right child of $v$. If $v$ is empty, it returns $\emptyset$. If the value of node $v$ equals the target $x$, it returns $v$ itself. This function could be seen as a search algorithm on a binary tree.

  • 1
    $\begingroup$ This seems quite correct. $\text{otherwise}$ is also $v\ne\emptyset\land g(v)=x$. $\endgroup$
    – user16034
    Jul 19, 2023 at 15:08
  • $\begingroup$ @YvesDaoust Thanks for confirming, will the notation be more confusing if I replace $\ell(v)$ and $r(v)$ with $v_{\ell}$ and $v_r$? $\endgroup$
    – ilovewt
    Jul 20, 2023 at 3:44
  • $\begingroup$ IMO, this is a bad idea. $\endgroup$
    – user16034
    Jul 25, 2023 at 19:07

1 Answer 1


It is correct, but it could be made more readable.

$$ search(node, val) = \begin{cases} \emptyset & \text{if } node = \emptyset \\ search(node.left, val) & \text{if } node.val > val \\ search(node.right, val) & \text{if } node.val < val \\ node & \text{otherwise} \end{cases} $$


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