1. How to Draw a Tree
L-Systems in Computer Graphics
Steven Janke

2. Natural Trees

3. Tree Shapes

4. Euclidean Geometry Approach

5. Self-Similarity of Organic Forms

6. Computation = Processing Strings
Input:
Output:
aababccacabb
adfeeefg
Computer
Interpretation:

7. Alphabet of characters.
L-System
(Named after biologist Astrid Lindemayer in 1970’s)
Alphabet of characters.
First string called the axiom.
Set of productions showing how to replace characters.
All appropriate productions applied at once.
Example:
Alphabet: {a, b}
Axiom: ab
Productions: a bab , b a
Derivation: ab baba abababab

8. Languages:
L(G) = set of strings that can be derived from the system G.
Example 1:
Axiom: ab Productions: a a b ab
ab aab aaab aaaab L(G) = { an b | n > 0 }
Example 2:
Axiom: a Productions: a b b ab
a b ab bab abbab bababbab
L(G) = ?

9. Languages:
L(G) = set of strings that can be derived from the system G.
Example 1:
Axiom: ab Productions: a a b ab
ab aab aaab aaaab L(G) = { an b | n > 0 }
Example 2:
Axiom: a Productions: a b b ab
a b ab bab abbab bababbab
L(G) = { s | s0 = a, s1 = b, sn = sn-2 sn-1 }

10. Turtle Interpretation:
Simple L-System:
Alphabet: { F, +, - }
Axiom: F-F-F-F
Production: F FF
Turtle Interpretation:
F means draw a line segment in current direction.
+ means turn left.
- means turn right.
Start
Initial direction
F-F-F-F means:
Delta = 90 degrees

11. Branching L-Systems: Add two characters to alphabet: [ and ]
Interpret [ to mean “start branch”.
Interpret ] to mean “end branch”.
F[+F][-F] means:
Initial Direction
Start
Delta = 45 degrees

12. Turtles in 3D: + = left turn - = right turn & = pitch down
Head Up
+ = left turn
- = right turn
& = pitch down
^ = pitch up
/ = roll right
\ = roll left

13. Growth Functions:
F(k) = length of the kth word in the derivation sequence.
Example:
a aa Axiom: a
F(k) = 2k
a abcc b bcc c c Axiom: a
a abcc abccbcccc abccbccccbcccccc
F(k) = k2 F(

14. Growth Functions: G: a b b ab Axiom: b
ak = number of a’s at iteration k.
bk = number of b’s at iteration k. ak bk
ak+1
bk+1 =
Theorem: Every growth function for an L-system is a linear combination
of terms that are polynomials times exponential functions.
Problem: Plants usually grow according to a logistic (or sigmoidal) function.

15. Parametric L-Systems:
Axiom: A(3)
A(x) : x< B(x+1)A(x*r)
B(y) : * F(y)[+F(y/2)][-F(y/2)]
Interpretation:
F(x) means draw a segment of length x.
+(x) means turn left x degrees.

18. Context Sensitive L-Systems
Axiom: SFFFFA
Production: SF FS SA B
SFFFFA FSFFFA FFSFFA FFFSFA FFFFSA FFFFB
Axiom: S[FA][FFA]
S[FA][FFA] [FSA][FSFA] [FB][FFSA] [FB][FFB]

19. Developmental model using signals:

21. L-System Extensions: Gravity - pull on branches.
Phyllotaxis - angle and position of branches.
Phototropism - towards the light.
Self-Organizing - branch into free spaces.
Implementation: At each iteration, interpret the string and then
decide based on the geometry and environment how to apply
productions for the next iteration.

22. Colonization Algorithm: (Runions, Lane, and Prusinkiewicz 2007)

23. Colonization Algorithm:

24. Self-Organization Algorithm: (Palubicki 2009)

25. Self-Organization Algorithm:

26. Equivalence: G1: a bb b a Axiom: b b a bb aa bbbb
G2: a b b aa Axiom: a
a b aa bb aaaa
L(G1) = L(G2)
Is there an algorithm for determining if two L-Systems are equivalent?

27. Connection between Languages and Machines
Recursively Enumerable
Context Sensitive
Context Free
Regular
L-Systems

34. Iterated Function System: