The Tree Evaluation Problem (TreeEval) is a computational problem originally proposed as a candidate to prove a separation between complexity classes P and L. Recently, this problem has gained significant attention after Cook and Mertz (STOC 2024) showed that TreeEval can be solved using $O(\log n\log\log n)$ bits of space. Their algorithm, despite getting very close to showing TreeEval $\in$ L, falls short, and in particular, it does not run in polynomial time.

In this work, we present the first polynomial-time, almost logarithmic-space algorithm for TreeEval. For any $\varepsilon>0$, our algorithm solves TreeEval in time $\mathrm{poly}(n)$ while using $O(\log^{1 +\varepsilon}n)$ space. Furthermore, our algorithm has the additional property that it requires only $O(\log n)$ bits of free space, and the rest can be catalytic space. Our approach is to trade off some (catalytic) space usage for a reduction in time complexity.