A separable permutation, which is generated by two elementary operations called direct sum and skew sum of permutations, is a permutation that avoids both the patterns 2413 and 3142. A di-sk tree is a rooted binary tree whose nodes are labeled by $\oplus$ or $\ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. In 2021, S. Fu, Z. Lin, and Y. Wang proved the equidistribution of some statistics concerning tree traversal and conjectured three symmetries hold. The main objective of this paper is to confirm this conjecture. This is accomplished both algebraically and combinatorially. In this paper, we obtain a closed form for the multivariate generating function of statistics concerning tree traversal. Besides, by a series of calculations using generating functions, we confirm this conjecture algebraically. Also, by transforming some statistics to the number of some specific nodes, we give a combinatorial proof of the first symmetry. Then by a generalization of a natural bijection between binary trees and plane trees, and the mirror symmetry, we give the combinatorial proof of the last two symmetries.