Tree and XOR

You are given a connected undirected graph without cycles (that is, a tree) of $$$n$$$ vertices, moreover, there is a non-negative integer written on every edge.

Consider all pairs of vertices $$$(v, u)$$$ (that is, there are exactly $$$n^2$$$ such pairs) and for each pair calculate the bitwise exclusive or (xor) of all integers on edges of the simple path between $$$v$$$ and $$$u$$$. If the path consists of one vertex only, then xor of all integers on edges of this path is equal to $$$0$$$.

Suppose we sorted the resulting $$$n^2$$$ values in non-decreasing order. You need to find the $$$k$$$-th of them.

The definition of xor is as follows.

Given two integers $$$x$$$ and $$$y$$$, consider their binary representations (possibly with leading zeros): $$$x_k \dots x_2 x_1 x_0$$$ and $$$y_k \dots y_2 y_1 y_0$$$ (where $$$k$$$ is any number so that all bits of $$$x$$$ and $$$y$$$ can be represented). Here, $$$x_i$$$ is the $$$i$$$-th bit of the number $$$x$$$ and $$$y_i$$$ is the $$$i$$$-th bit of the number $$$y$$$. Let $$$r = x \oplus y$$$ be the result of the xor operation of $$$x$$$ and $$$y$$$. Then $$$r$$$ is defined as $$$r_k \dots r_2 r_1 r_0$$$ where:

$$$$$$ r_i = \left\{ \begin{aligned} 1, ~ \text{if} ~ x_i \ne y_i \\ 0, ~ \text{if} ~ x_i = y_i \end{aligned} \right. $$$$$$