We extend the theory of minimal absent words to (rooted and unrooted) trees, having edges labeled by letters from an alphabet $Σ$ of cardinality $σ$. We show that the set $\text{MAW}(T)$ of minimal absent words of a rooted (resp. unrooted) tree $T$ with $n$ nodes has cardinality $O(nσ)$ (resp. $O(n^{2}σ)$), and we show that these bounds are realized. Then, we exhibit algorithms to compute all minimal absent words in a rooted (resp. unrooted) tree in output-sensitive time $O(n+|\text{MAW}(T)|)$ (resp. $O(n^{2}+|\text{MAW}(T)|)$ assuming an integer alphabet of size polynomial in $n$.