A semilattice tree T with 0 is slim if there is a chain C with 0 so that the lattices θ (T) and θ(C) of semilattice congruences are isomorphic. This paper establishes elementary consequences of slimness and uses simple constructive techniques to show certain small trees slim. If T is the union of at most countably many branches, each of which has a maximum or a countable cotinal subset, then T is slim. For trees with enough maximals slimness is equivalent with not having any uncountable anti-chains. If a tree T has a countable cofinal subset then T is slim. Thus finitary trees are slim.
