a random talk on random trees
DESCRIPTION
This is a talk I gave the 8 Nov 2011 for the postgraduate seminar at the University of Bath. The talk was roughly half an hour long and it's a small introduction to my MSc dissertation.TRANSCRIPT
Horacio González Duhart [email protected]
What is a tree?
root
leaf
branch
What is a tree?
root
branch
leaf
What is a tree?
root
branch
leaf
What is a tree?
root
branch
leaf
node
children
father
What is a tree?
“A tree is a connected graph which has no cycles.”
What is a random tree?
What is a random tree?
1. Start with the root.
What is a random tree?
1. Start with the root. 2. Throw the dice!
What is a random tree?
1. Start with the root. 2. Throw the dice! 3. Put as many children as the result
of the dice.
What is a random tree?
1. Start with the root. 2. Throw the dice! 3. Put as many children as the result
of the dice. 4. Repeat from step 1 for each of the
new children. (iid)
Throwing the dice… or offspring distribution
Let L be the random variable that represents the offspring distribution. Which can be summarised in a table like this:
0
1
2
n
Let the maths begin!!!
Define
Then we need that
Let the maths begin!!!
Define
Then we need that
Some examples:
Let the maths begin!!!
Define
Then we need that
Now, notice that for
Let the maths begin!!!
Define
Then we need that
Now, notice that for
Generating function (of L evaluated at s)
More properties…
is continuous…
More properties…
is continuous…
It is increasing…
More properties…
is continuous…
It is increasing…
It is convex.
More properties…
is continuous…
It is increasing…
It is convex.
More properties…
is continuous…
It is increasing…
It is convex…
It looks either like this:
or
Weren’t we talking about trees?
Back on the tree…
The n-th generation
Let be the random variable that denotes the size of the n-th generation.
What is the distribution of ?
What is the distribution of ?
First, notice that
What is the distribution of ?
First, notice that
And calculate the generating function…
What is the distribution of ?
First, notice that
And calculate the generating function… which is easy to find by induction to:
Some interesting questions…
Some interesting questions…
What is the probability that the tree becomes extinct?
Some interesting questions…
What is the probability that the tree becomes extinct?
Some interesting questions…
What is the probability that the tree becomes extinct?
Man! That looks hard! Isn’t there an easier way?
Some interesting questions…
What is the probability that the tree becomes extinct?
Man! That looks hard! Isn’t there an easier way?
Some interesting questions…
Under what conditions will the tree become extinct with probability 1?
What is the probability that the tree becomes extinct?
Man! That looks hard! Isn’t there an easier way?
Some interesting questions…
Under what conditions will the tree become extinct with probability 1?
What is the probability that the tree becomes extinct?
Man! That looks hard! Isn’t there an easier way?
There are almost surely more interesting questions on Galton-Watson trees, but I guess this is enough for now and time is running out… Cheers! … and enjoy the cake!