We study the problem of collective tree exploration in which a team of $k$ mobile agents must collectively visit all nodes of an unknown tree in as few moves as possible. The agents all start from the root and discover adjacent edges as they progress in the tree. Communication is distributed in the sense that agents share information by reading and writing on whiteboards located at all nodes. Movements are asynchronous, in the sense that the speeds of all agents are controlled by an adversary at all times. All previous competitive guarantees for collective tree exploration are either distributed but synchronous, or asynchronous but centralized. In contrast, we present a distributed asynchronous algorithm that explores any tree of $n$ nodes and depth $D$ in at most $2n+O(k^2 2^kD)$ moves, i.e., with a regret that is linear in $D$, and a variant algorithm with a guarantee in $O(k/\log k)(n+kD)$, i.e., with a competitive ratio in $O(k/\log k)$. We note that our regret guarantee is asymptotically optimal (i.e., $1$-competitive) from the perspective of average-case complexity. We then present a new general lower bound on the competitive ratio of asynchronous collective tree exploration, in $Ω(\log^2 k)$. This lower bound applies to both the distributed and centralized settings, and improves upon the previous lower bound in $Ω(\log k)$.