2. (12 points) A particular tree grows in the following way: It starts as a single bud. . Each week, all the exposed buds either flower or branch. The probability that a bud will flower is 3/5. The probability that a bud will branch is 2/5. 2/5 3/5 . If the bud branches, then it splits into two branches each with a single exposed bud at the end. A mature tree is a tree that does not have any exposed buds. A mature tree does not grow any more. A scrub is a mature tree that never branches. A scrub is a mature tree with only one flower. Here is an example of a mature tree that is not a scrub: You are farming these trees for their flowers and you grow n trees in a circle. Scrubs are not profitable so you cannot sell them. Other non-scrub flowers will sell for 1 dollar each. (a) If you plant n trees, what is the expected number of non-scrub flowers? (b) Scrubs are not profitable so you use a lawn mower to destroy them one run at a time. A run of scrubs is a consecutive sequence of scrubs that can't be extended in either direction. (Consider a field of all scrubs to be 0 runs.) For example: these 15 mature trees pictured below have 4 runs of scrubs: What is the expected number of runs of scrubs in terms of n? 2. (12 points) A particular tree grows in the following way: It starts as a single bud. . Each week, all the exposed buds either flower or branch. The probability that a bud will flower is 3/5. The probability that a bud will branch is 2/5. 2/5 3/5 . If the bud branches, then it splits into two branches each with a single exposed bud at the end. A mature tree is a tree that does not have any exposed buds. A mature tree does not grow any more. A scrub is a mature tree that never branches. A scrub is a mature tree with only one flower. Here is an example of a mature tree that is not a scrub: You are farming these trees for their flowers and you grow n trees in a circle. Scrubs are not profitable so you cannot sell them. Other non-scrub flowers will sell for 1 dollar each. (a) If you plant n trees, what is the expected number of non-scrub flowers? (b) Scrubs are not profitable so you use a lawn mower to destroy them one run at a time. A run of scrubs is a consecutive sequence of scrubs that can't be extended in either direction. (Consider a field of all scrubs to be 0 runs.) For example: these 15 mature trees pictured below have 4 runs of scrubs: What is the expected number of runs of scrubs in terms of n