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The degree sequence $(N_{i,j}(k),1\leq i,j\leq d,k\geq 0)$ of a multitypeforest with $d$ types, is the number of individuals type $i$, having $k$children type $j$. We construct a multitype forest sampled uniformly from allmultitype forest with a given degree sequence (MFGDS). For this, we use anextension of the Ballot Theorem by (Chaumont and Liu, 2016), and generalize theVervaat transform (Vervaat, 1979) to multidimensional discrete exchangeableincrement processes. We prove that MFGDS are extensions of multitypeGalton-Watson (MGW) forests, since mixing the laws of the former, one obtainsMGW forests with fixed sizes by type (CMGW). We also obtain the law of thetotal population by types in a MGW forest, generalizing Otter-Dwass formula(Otter 1949, Dwass 1969). We apply this to obtain enumerations of plane,labeled and binary multitype forests having fixed roots and individuals bytypes. We give an algorithm to simulate certain CMGW forests, generalizing theunitype case of (Devroye, 2012).

On Multitype Random Forests with a Given Degree Sequence, the Total Population of Branching Forests and Enumerations of Multitype Forests