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This article is dedicated to the study of asymptotically rigid mapping classgroups of infinitely-punctured surfaces obtained by thickening planar trees.Such groups include the braided Ptolemy-Thompson groups $T^\sharp,T^\ast$introduced by Funar and Kapoudjian, and the braided Houghton groups$\mathrm{br}H_n$ introduced by Degenhardt. We present an elementaryconstruction of a contractible cube complex, on which these groups act withcube-stabilisers isomorphic to finite extensions of braid groups. As anapplication, we prove Funar-Kapoudjian's and Degenhardt's conjectures byshowing that $T^\sharp,T^\ast$ are of type $F_\infty$ and that $\mathrm{br}H_n$is of type $F_{n-1}$ but not of type $F_n$.

Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson's and Houghton's groups