Day 19, Mat385

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Announcements
- The Zooms continue
  A menu of our past zooms...
- All but 2 projects have been green-lighted. Time to get at those! I'm going to be adding you all to the wiki at some point. I'll alert you when I do, because your links to establish your accounts (automagically generated) will almost certainly go to spam....:(
- 6.2 Homework graded: 2, 36, 56
  - #36: shoot for some sort of optimization, if possible:
  - #56: how many non-isomorphic rooted trees with $n$ nodes?
    1. Find some way to organize your efforts (many missed 1; some even 2 or more...:):
    2. Another organizational scheme is by the number of children $r$ has (as you can see, there are four of one child, three of two children; one of three and one of four.
    3. Another strategy (and the best strategy!) is recursive:
      How do we build trees of five nodes? Out of trees of fewer nodes!
      1. one (rooted) tree with one node
      2. one tree with two nodes
      3. two trees with three nodes
      4. four trees with four nodes
      5. nine trees with five nodes
      Take a moment and think about how to build all the trees with five nodes out of the trees with fewer nodes.
      How many non-isomorphic trees are there with six nodes?
- Our second exam is next week, Thursday 4/7. On Tuesday we'll review. I've promised some review problems by Monday -- hopefully sooner.
  The exam will cover everything from
  1. 2.2: Induction
  2. 3.1: Recursive Definitions
  3. 3.2: Recurrence Relations
  4. 3.3: Analysis of Algorithms
  5. 4.1: Sets
  6. 6.1: Graphs
  7. 6.2: Trees
  That should be enough for this exam. I'll hold off on 6.3, and following. The final will be in two parts: new stuff (6.3 and on), and the comprehensive part.
  Here is an overview of the sections above.
Last time:
- We nearly finished Section 6.3 (Decision Trees), but we have a few more bits to do.
Today:
- Today we'll finish Section 6.3 (Decision Trees).
- Then we'll get going on Euler Paths and Hamiltonian Circuits.
Links:
- Rooted Trees
- Kaliningrad, Russia (formerly Konigsberg)
- From the 7.2 highlights:
- EulerPath:
- Graph Terminology Handout
- Tree Terminology Handout
- If you check out this code, you'll see where I compute the power set ("all-subsets") of a set, to cover all the bases. I think that the end result is pretty beautiful, too!
- Recursive in-order, post-order, pre-order in lisp
- Recursive computation of Fibonacci numbers (in lisp)
- Recursive computation of rows from Pascal's triangle (in lisp)
- "String exponention" (in lisp)
- Here is the recursive selection sort algorithm, in lisp:
- Here is a recursive binary search algorithm in lisp:
- Closed-form computation of Fibonacci numbers (in lisp)
- Some examples of the Euclidean algorithm:
  1. Simple and clean recursive Euclidean Algorithm in lisp:
  2. Written as a while loop, using the recurrence relation:
- Here is a recursive bubbleSort algorithm in lisp
  
  (coded up following our author's algorithm, in section 3.3, p. 214, as closely as possible).

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