Binary Trees and Pascal’s Triangle

We tried a new Italian restaurant last night.  It used paper to cover the tables and they let kids draw on the table covers with crayons which was a nice way to pass the time.  I was surprised to see that my younger son was drawing binary trees.  He said that he remembered them from an old Vi Hart video, which was a little strange since it is a Thangsgiving video.  Oh well, no telling what kids will remember:

I had a different project on tap for today’s Family Math, but when your kid is drawing binary trees on the table it is probably a sign, so plans changed!  We started our talk this morning with a quick review of what binary trees are, and then talked about a few simple properties that they have:

Next we build on the topic that we touched at the end of the last video – representing coin flips in a binary tree.  If we want to keep track of only the number of heads and tails that we’ve seen, some sequences that we’ve seen before make a surprising appearance in our little tree:

Next we moved on to showing a picture of how the binary tree can merge into Pascal’s triangle.  It was neat to see that the kids had seen how the “diamonds” would appear.  We also talked a little informally about why the pattern here is indeed the same as in Pascal’s triangle.  One of the other fun things we look at in this video is how the row sums that were easy to see in the binary tree carry over to this setting:

Finally, I wanted to show how this idea could help us solve a problem that they’d not seen before (though a pretty standard Pascal’s triangle problem).  The problem ask about counting different paths in a lattice.  We can think of the go right / go up choice as similar to the heads / tails coince from the binary tree example:

So,  a little doodling on our restaurant table cloth last night turned into a fun little Family Math talk.  Always fun to see what kids remember from things that they’ve seen (and when they remember them, too, I suppose!).