How can I get wood in a desert?

Officially, you can't. Just run in one direction till you find another biome with trees.

Assuming you play this as a challenge - no other biomes, just surviving in the desert:

As of 1.9 dead bushes when broken drop sticks. This makes the challenge much easier than before.

Seek caves (may take some trial and error; some may be buried under the sand) and try to find an abandoned mine. There's wood aplenty in there, in the form of the supports. Some smart creeper management can earn you cobble and coal. Some village houses contain crafting benches you can then use to assemble stone tools without access to wood.

Edit:

Desert hell was a challenge stream of gameplay in a desert world, with no mineshafts, villages, and caverns. You may watch it for some very creative solutions:

• Wood is still obtainable from Strongholds - but locating one takes enormous effort.
• Dead bushes and witches are important sources of sticks. These are a very important resource.
• Cacti are good as a makeshift weapon if you lure the mobs to walk onto them.
• Suffocating mobs is a good way of killing them; there are some ways demonstrated to do this.
• Temples and Dungeons were very important - especially allowing you to get a bucket. Bucket of lava is a mighty weapon.
• Using spider webs you can create wool, which then you can set on fire using lava, to start a nether portal (which you can cast from lava and water obtained from a desert well).

While it is true that there are no trees in desert biomes, I would recommend you find the nearest biome that actually has trees; although there are some workarounds.

If you are lucky enough, you may be able to stumble across an NPC village, or a Desert Temple; which may contain actuall chests and or wood to provide for manually making them.