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A solution in Python, with some comments.
# Set the initial height of the tree
height = 1
# Let's setup a range that'll cover every cycle and iterate through it.
for cycle in range(0, n + 1):
# Nothing happens during the first cycle, so we skip the rest of the loop.
if cycle == 0: continue
# Odd cycles are 'spring', so we add height to itself, otherwise add 1.
height += height if cycle % 2 == 1 else 1