Thanks for the comments, everyone. I did make one mistake however (based on what I read earlier.) Every website I checked said that Cesar Chavez Street was originally named First Street when the city was laid out in 1839. Apparently, that is not true. It was named Water Ave.
The main boundaries in 1839 were North Avenue (now 15th Street); East Avenue (now I H 35): Water Avenue (now Cesar Chavez) and West Avenue, which is still called by that name today.
An interesting article I found over at the Austin Chronicle states that the tree-named streets extended all the way to 28th street by the time they were changed to numbers in 1884.
By Any Other Name
The Names of Austin's Cherished Landmarks Are Half History, Half Serendipity
I could only find the tree streets (and three exceptions) up as far as 19th Street.
Current Name/Old Name
Willow Street - Willow Street
Cesar Chavez - Water Avenue
2nd Street - Live Oak Street
3rd Street - Cypress Street
4th Street - Cedar Street
5th Street - Pine Street
6th Street - Pecan Street
7th Street - Bois d'Arc Street
8th Street - Hickory Street
9th Street - Ash Street
10th Street - Mulberry Street
11th Street - Mesquite Street
12th Street - College Avenue
13th Street - Peach Street
14th Street - Walnut Street
15th Street - North Avenue
16th Street - Cherry Street
17th Street - Linden Street
18th Street - Chestnut Street
MLK (19th St) - Magnolia Street
Here's links to some cool old (big) street maps of Austin
Austin Undated
Austin 1890-1895
On the 1890-1895 map, when you zoom in, follow College Avenue out to the west and you'll see the old TMI Castle which overlooks Lamar Blvd. today.
Also, comparing the undated map to the 1890-1895 map, you can really see how Austin was growing back then.