  We study the problem of collective tree exploration (CTE) where a team of $k$
agents is tasked to traverse all the edges of an unknown tree as fast as
possible, assuming complete communication between the agents. In this paper, we
present an algorithm performing collective tree exploration in only
$2n/k+O(kD)$ rounds, where $n$ is the number of nodes in the tree, and $D$ is
the tree depth. This leads to a competitive ratio of $O(\sqrt{k})$ for
collective tree exploration, the first polynomial improvement over the initial
$O(k/\log(k))$ ratio of [FGKP06]. Our analysis relies on a game with robots at
the leaves of a continuously growing tree, which is presented in a similar
manner as the `evolving tree game' of [BCR22], though its analysis and
applications differ significantly. This game extends the `tree-mining game'
(TM) of [Cos23] and leads to guarantees for an asynchronous extension of
collective tree exploration (ACTE). Another surprising consequence of our
results is the existence of algorithms $\{A_k\}_{k\in \mathbb{N}}$ for layered
tree traversal (LTT) with cost at most $2L/k+O(kD)$, where $L$ is the sum of
edge lengths and $D$ is the tree depth. For the case of layered trees of width
$w$ and unit edge lengths, our guarantee is thus in $O(\sqrt{w}D)$.
