NormalizationConstant[TreeList_] := 
  Module[{tree, uses, offered, saturated, recursion, flows, nfl, isets, rt, tt, g, G, a}, 
    (* Definitions *) 
  tree[n_] := {Flatten[{n, (tree[#][[1]] & /@ Flatten[ Position[TreeList, {n, __}]])}], TreeList[[n]][[2]],
        tree /@ Flatten[ Position[TreeList, {n, __}]]} ;
  offered[t_, ind_] :=  Min[t[[2]], 
        If[MemberQ[ind, t[[1, 1]] ], Infinity , 0] + 
          Plus @@ (offered[#, ind] & /@ Last[t])];
  saturated[{}, ind_] := {Null, Null, False};
  saturated[t_, ind_] :=  If[offered[t, ind] == t[[2]], {Intersection[ind, t[[1]]], t[[2]], True},
         If[(i =  Position[Last[saturated[#, ind]] & /@ Last[t],  True]) == {}, {Null, Null, False},
  saturated[Last[t][[i // First // First]], ind] ]];
  recursion[ind_] := Module[{sat, ii, iii, r, gg, c},
        {ii, c} = Drop[saturated[tt, ind], -1];
        iii = Complement[ind, {#}] & /@ ii;
        gg = g /@ iii; 
        hh = h /@ iii;
        r = Last /@ TreeList[[ii]] ;
        c -= Plus @@ r;
        gg = r.gg/c ];
    (* Initializations *);
  flows = Position[TreeList, {_, _, _}] // Flatten; nfl = Length[flows];
  isets = Table[KSubsets[flows, k], {k, nfl}] // Flatten[#, 1] &;
  rt = Position[TreeList, {Null, __}] // First // First;
    tt = tree[rt];
    (* Actual recursion *)
    G = g[{}] = 1;
    G += (g[#] = recursion[#]) & /@ isets;
  G // Simplify]