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CSE 491 Introduction to Bioinformatics

创建日期：2024-08-01 16:46:07

所属分类：计算机computer science

标签： CSE 491

Introduction to Bioinformatics

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CSE 491 Introduction to Bioinformatics

Homework 2

Type-set solutions are preferred. Remember that written homework solutions must be type-set

to be eligible for electronic submission. If you submit a hand-written solution, please make an

effort to use neat, clean handwriting. Hand-written homework solutions must physically submitted

in-person. See syllabus for more details regarding this policy.

Reminder: CSE 491 adheres to Michigan State University’s policies on academic integrity. No

collaboration is allowed on homeworks, and all work must be your own.

1. (10 points) The unrooted version of a rooted leaf-labeled binary tree can be obtained by

simply ignoring edge directionality, deleting the root, and then “connecting” its two incident

edges.

Equivalently, an unrooted leaf-labeled binary tree is an undirected graph G = (V, E) where

G is connected, acyclic, all nodes v ∈ V have degree of either 1 (i.e., leaf nodes) or 3 (i.e.,

internal nodes), and all leaf nodes have distinct labels. Recall that the degree of a node in

an undirected graph is defined as the number of edges incident upon it.

For example, see Figure 1 which shows three distinct unrooted leaf-labeled binary trees. These

are the only possible distinct unrooted leaf-labeled binary trees with four taxa.

Using induction, prove that the number of distinct unrooted leaf-labeled binary trees on n

taxa is (2n 5)!!.

Figure 1: These are the only possible distinct unrooted leaf-labeled binary trees with four taxa.

(5 points) Without using induction, prove that the number of distinct unrooted leaf-labeled

binary trees on n taxa is (2n 5)!!.

(Hint: you can make use of the fact that the number of distinct rooted leaf-labeled binary

trees on n taxa is (2n 3)!!, as we proved in class.)

2. (5 points) Let i be a natural number. Let T be the set of rooted binary trees which have

n = 2i

leaves. Let X T be the subset of such trees having maximal height. What property

(or properties) do the trees in X have in common? Prove that your answer is correct.

(5 points) Let Y T be the subset of such trees having minimal height. What property (or

properties) do the trees in Y have in common? Prove that your answer is correct.

3. Optional bonus problem (+10 extra credit points):

Recall that the nth Fibonacci number can be defined using a recursive function f:

f(0) = 1

f(1) = 1

f(n) = f(n1) + f(n2) where n is a natural number greater than 1

Also recall that the golden ratio constant is defined to be, and the complement of

φ is defined to be ψ = 1

Prove that f can be equivalently expressed in the following non-recursive form:

f(n) = φ

where n is a natural number

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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