Error in Recursive Routine

error.rb

"I need to find the set of possible ways of traversing a tree of height n. For a tree of height 4, that looks like this: "

h4 = [
  [0, 0, 0, 0],
  [0, 0, 0, 1],
  [0, 0, 1, 1],
  [0, 0, 1, 2],

  [0, 1, 1, 1],
  [0, 1, 1, 2],
  [0, 1, 2, 2],
  [0, 1, 2, 3]
]

"Generally, I can initialize this set of sets, with everything at zero, like so: "

$tree_height = 4 

$matrix = []

($tree_height*2).times do |rows|
  row = []
  $tree_height.times do |columns|
    row << 0 
  end 
  $matrix << row 
end 


"My logic is, I can then make this the set of all possible traversals by repeatedly halving the rows, and, then, let's say to the first half, adding one to all columns after and including x, starting at one, and each time increment by one."
 
h4 = [
  [0, 1, 1, 2], # first halving, x = 1
  [0, 1, 1, 1],
  [0, 1, 2, 3],
  [0, 1, 2, 2],
 
  [0, 0, 1, 2], # second halving, x = 2
  [0, 0, 1, 1],
 
  [0, 0, 0, 1], # third having, x = 3
 
  [0, 0, 0, 0]
]

"So I need a recursive routine, which I have written as such:"

def add_one_after_x_for_rows x, a, b
  (a..b-1).each do |row|
    (x+1..$tree_height-1).each do |column|
      $matrix[row][column] += 1 
    end 
  end 
end 

def assemble_rows_for_columns_after a, b, x
  add_one_after_x_for_rows(x, a, b/2) 
  if x < $tree_height 
    assemble_rows_for_columns_after(a,b/2,x+1)
    assemble_rows_for_columns_after(b/2,b,x+1)
  end 
end 

assemble_rows_for_columns_after(0,$tree_height*2, 0)

"But this is wrong! Its output looks more like:"
 
output = [
        [0, 1, 2, 3],
        [0, 1, 2, 2],
 
        [0, 1, 1, 1],
        [0, 1, 1, 1],
 
        [0, 0, 0, 0],
        [0, 0, 0, 0],
 
        [0, 0, 0, 0],
        [0, 0, 0, 0]
]

"What is the problem with my routine?"