Monday, September 22, 2008
Ternary Trees
So...I spent most of the weekend mulling over this problem. So far, I've probably spent 10-12 hours concentrated time and more than a week thinking about it off and on. I've drawn the tree sets for n=3, n=4 and n=5. Still no pattern has emerged that I can spot. At this point, it would be an understatement to say that I'm discouraged. More than that, I need to stop working on this and get on with other things. However, assuming that there is a formula, say f(n) that produces the number of non-equivalent trees (I'd rather say unique -- easier to say and understand), I can produce a proof easily enough. For the moment, I'll assume that I have such a formula. Who knows? Maybe one will show up before I hand it it.
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10-12 hours is too much time to spend without seeing a TA or the instructor. I doubt that the numbers themselves suggest a pattern (they certainly don't to me).
More helpful is to think about what system you use for arranging your drawings of trees. Do you group them, or enumerate them, in a way that helps you decide you've gotten all instances, and haven't double-counted. The idea is that your system for systematically listing trees can be expressed as a formula.
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