Staring at trees...

I am sorry to admit that since the ternary tree problem was postponed to the next assignment, I had not pondered upon it. Now, when my stream of consciousness seems to be ubiquitously defined by a common cold virus, it seems like it could have been a good idea to read the hand out properly a week ago, in the heights of health. But at least I can still plan the steps to be taken over the next few days. Based on Polya's memory-tickling problem-solving approaches, I intend to:

1) Draw trees - ternary trees of n nodes. I have heard of cases where hundreds of tree sketches were needed. I will hopefully stop before my first hundred, if I get there, and approach the problem from an alternative perspective then. Careful tracking of the number of nodes and the number of non-equivalent ternary trees will be crucial at this stage. Perhaps considering the subtrees of the subtrees will help... we'll see.

2) Visual quantification - I usually find patterns more appealing when they can be perceived in some way. It's like unwinding a function - aligning columns of expressions can often make clear what the numerical pattern for a particular case is.

3) Small n results - attempting to find patterns for the lower values of n can often be confusing as a pattern does not always arise immediately, but it can be very valuable to observe how the pattern progresses as the trees bloom deeper and depper.

4) Detecting a pattern!

5) Verifying pattern - for other values of n that are smaller (including, perhaps, a base case or a few base cases) and for values of n that are bigger.

6a) Discard pattern - if the perceived pattern did not work, check steps from number 1) to see if anything went wrong at any stage e.g. trees were not properly drawn, trees to be considered should have been less/more, etc.

6b) Prove pattern - maybe through some flavour of induction?

I will report back once (if?) I figure this problem out. Also note that the problem set 4 might be useful for the 4th problem of the assignment, as it seems to be a smaller version of a similar exercise. I will write about this guess on the next entry as well.