At the end of each growing season, Joy likes to prune dead leaves
from her favourite tree. She does this by cutting branches. For this
tree, shown below, there are \(15\)
leaves she wants to remove. She decides to give an approximate time it
will take to cut each branch. These times are shown for each branch.
The tree starts by dividing into three main branches: left, centre, and right.
The main left branch has time \(15\) and leads to the three leftmost leaves
as follows: This branch divides into three branches, two with time \(5\) and one with time \(6\), each ending with a leaf.
The main centre branch has time \(25\) and leads to the eight centre leaves
as follows: This branch divides into two branches: left centre and right centre.
The left centre branch has time \(10\) and has two branches of its own:
The left one has time \(6\) and leads to branches with times \(2\) and \(3\), each ending with a leaf.
The right one has time \(6\) and leads to branches with times \(2\) and \(5\), each ending with a leaf.
The right centre branch has time \(11\) and has two branches of its own:
The left one has time \(2\) and ends with a leaf.
The right one has time \(8\) and leads to a branch with time \(4\) ending with a leaf and a branch with
time \(6\) which leads to branches with
times \(3\) and \(2\), each ending with a leaf.
The main right branch has time \(9\) and leads to the four rightmost leaves
as follows: This branch divides into a branch with time \(3\) ending with a leaf, a branch with time
\(1\) ending with a leaf, and a branch
with time \(5\) which leads to branches
with times \(1\) and \(3\), each ending with a leaf.
When a branch is cut, all branches and leaves attached to it are
removed from the tree. For example, if you cut the branch labelled with
\(15\), the three leftmost leaves will
be removed.
What is the shortest amount of time in which Joy can remove all \(15\) leaves?
