Waste less time on Facebook — follow Brilliant.
×

Fibonacci Investigation

A couple of months ago, we were studying graphs in Further Maths (not an \({xy}\) plane, rather a series of nodes and edges), when we decided to do a little investigation with \({tree}\) \({diagrams}\). The question goes like this: what are number of \({distinct}\) \({tree}\) \({diagrams}\) with \({n}\) nodes? Well we found that if a graph has (2,3,4,5,6,7) nodes then it would have (1,1,2,3,5,8) distinct tree diagrams respectively, which follows the Fibonacci sequence. We were not able to prove why, so could anybody be able to provide a proof? (I have a feeling it might be to do with the number of distinct ways you can write 2[\({n}\)-1] as a sum of \({n}\) numbers, if \({n}\) = number of nodes)

Note by Curtis Clement
2 years, 11 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

I count 6 trees with 6 nodes up to graph isomorphism.

Lee Gao - 2 years, 11 months ago

Log in to reply

Check the order of your nodes - we didn't find any counter-examples as a class.

Curtis Clement - 2 years, 11 months ago

Log in to reply

OEIS A000055 agrees as well, the sequence proceed as 1, 1, 2, 3, 6, 11, 23, 47,...

Lee Gao - 2 years, 11 months ago

Log in to reply

@Lee Gao What is OEIS A000055 ?

Curtis Clement - 2 years, 11 months ago

Log in to reply

Log in to reply

A series of nodes and edges such that the graph is connected and there are no cycles. Also, the nodes have to be connected using the least number of nodes. Examples: (order of each node) 2 nodes = {1,1} 3nodes = {2,1,1} 4 nodes = {1,2,2,1} and {3,1,1,1} 5 nodes = (4,1,1,1,1}, {3,2,1,1,1} and {2,2,2,1,1} etc

Curtis Clement - 2 years, 11 months ago

Log in to reply

Can you define more precisely what you mean by "tree diagrams" ? It is known that (Cayley's formula) number of distinct labelled trees on \(n\) vertices is \(n^{n-2}\).

Abhishek Sinha - 2 years, 11 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...