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The Furca Fractalis tree grows in a very special way. Starting with the trunk there are three possibilities to continue growing:

  1. It can split in two branches.
  2. It can grow one branch and one leaf.
  3. It can end with two leaves.

Each unterminated branch again grows either of the three possibilities. In the final configuration all branch ends must be terminated by two leaves. The image above shows a valid example.

These trees are very rare – I'm very proud of having one in my garden. Apothecaries pay very well for its leaves if you collect them directly after they fell off in autumn. So I always collect them and make sure not to miss out a single leaf.

This year I collected 1583 leaves. How many branch segments (including trunk) does my tree have?

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  • $\begingroup$ +1 for the picture, and for the surprisingly elegant solution which for some reason I completely missed. $\endgroup$
    – Xenocacia
    Nov 3, 2017 at 2:37

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Your tree has

1582 branch segments

because

this number is always 1 less than the number of leaves.

One way to see this:

You can make any tree as follows. Start with trunk + two leaves. (Notice that this obeys the rule above.) Now repeatedly grow by replacing a leaf with a branch segment and two leaves on the end. (Notice that this can't "break" the rule above.)

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  • $\begingroup$ Ninja'd me :P $ $ $\endgroup$
    – boboquack
    Nov 3, 2017 at 0:10
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    $\begingroup$ Like number of matches in a knock-out tournament where there is a single overall winner. Charming story and picture too $\endgroup$
    – Laska
    Nov 3, 2017 at 1:14

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