What is the number of rooted planar decreasing trees on n vertices? By planar, I mean embedded in the plane (I think sometimes the terms, ordered and plane, are used interchangeably with planar).
Assume the vertices are labeled 1,2,...,n.  By decreasing, I mean the labels on the vertices along a path from the root to any other vertex will decrease.  Note that this implies the root is labeled with the integer n. 
The journal article I am reading states that there are (2n - 3)*(2n - 1)***(3)*(1) such trees.  But the article does not give a derivation.
Can someone explain how to count these trees?  

T[n_] := T[n] = 
  Total[Map[Apply[Multinomial, #] Product[T[p], {p, #}] &, 
    Level[Map[Permutations[#] &, Partitions[n - 1]], {2}]]]; Table[
 T[n], {n, 1, 20}]
1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075, \
13749310575, 316234143225,...
nn = 6; A[z_] := Sum[T[n] z^n/n!, {n, 1, nn}]; 
Range[0, nn]! CoefficientList[
   Series[1/(1 - u A[z]), {z, 0, nn}], {z, u}] // Grid
{"1", "0", "0", "0", "0", "0", "0"},
{"0", "1", "0", "0", "0", "0", "0"},
{"0", "1", "2", "0", "0", "0", "0"},
{"0", "3", "6", "6", "0", "0", "0"},
{"0", "15", "30", "36", "24", "0", "0"},
{"0", "105", "210", "270", "240", "120", "0"},
{"0", "945", "1890", "2520", "2520", "1800", "720"}
