The Tree Evaluation Problem ($TreeEval$) (Cook et al. 2009) is a central candidate for separating polynomial time ($P$) from logarithmic space ($L$) via composition. While space lower bounds of $\Omega(\log^2 n)$ are known for multiple restricted models, it was recently shown by Cook and Mertz (2020) that TreeEval can be solved in space $O(\log^2 n/ \log \log n)$. Thus its status as a candidate hard problem for $L$ remains a mystery.

Our main result is to improve the space complexity of $TreeEval$ to $O(\log n \cdot \log \log n)$, thus greatly strengthening the case that Tree Evaluation is in fact in $L$.

We show two consequences of these results. First, we show that the KRW conjecture (Karchmer, Raz, and Wigderson 1995) implies $L \neq NC^1$; this itself would have many implications, such as branching programs not being efficiently simulable by formulas. Our second consequence is to increase our understanding of amortized branching programs, also known as catalytic branching programs; we show that every function $f$ on $n$ bits can be computed by such a program of length poly$(n)$ and width $2^{O(n)}$.