Let $T$ be an infinite countable labelled transition system with the shape of a tree and let $\sim$ be its similarity relation among nodes. Note that the alphabet of the labels is finite (hence the tree has finite branching). Let $T/\sim$ be its quotient w.r.t. the similarity relation, i.e. two nodes in $T$ are identified if they are similar. In general $T/\sim$ is a graph.

Is $T/\sim$ finite? If not in general, when is it finite?



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