We consider the problem of collaborative tree exploration posed by Fraigniaud, Gasieniec, Kowalski, and Pelc where a team of $k$ agents is tasked to collectively go through all the edges of an unknown tree as fast as possible. Denoting by $n$ the total number of nodes and by $D$ the tree depth, the $\mathcal{O}(n/\log(k)+D)$ algorithm of Fraigniaud et al. achieves the best-known competitive ratio with respect to the cost of offline exploration which is $Θ(\max{\{2n/k,2D\}})$. Brass, Cabrera-Mora, Gasparri, and Xiao consider an alternative performance criterion, namely the additive overhead with respect to $2n/k$, and obtain a $2n/k+\mathcal{O}((D+k)^k)$ runtime guarantee. In this paper, we introduce `Breadth-First Depth-Next' (BFDN), a novel and simple algorithm that performs collaborative tree exploration in time $2n/k+\mathcal{O}(D^2\log(k))$, thus outperforming Brass et al. for all values of $(n,D)$ and being order-optimal for all trees with depth $D=o_k(\sqrt{n})$. Moreover, a recent result from Disser et al. implies that no exploration algorithm can achieve a $2n/k+\mathcal{O}(D^{2-ε})$ runtime guarantee. The dependency in $D^2$ of our bound is in this sense optimal. The proof of our result crucially relies on the analysis of an associated two-player game. We extend the guarantees of BFDN to: scenarios with limited memory and communication, adversarial setups where robots can be blocked, and exploration of classes of non-tree graphs. Finally, we provide a recursive version of BFDN with a runtime of $\mathcal{O}_\ell(n/k^{1/\ell}+\log(k) D^{1+1/\ell})$ for parameter $\ell\ge 1$, thereby improving performance for trees with large depth.