The dictionary doesn't say that the sentence would be wrong if the word the was used. In fact, I think all four of the following sentences are grammatical and sensible, although they each conjure up slightly different images in my mind:
A: There was an accident here yesterday. A car hit a tree and the driver was killed.
B: There was an accident here yesterday. The car hit a tree and the driver was killed.
C: There was an accident here yesterday. A car hit the tree and the driver was killed.
D: There was an accident here yesterday. The car hit the tree and the driver was killed.
Version A (the original) is correct because the first sentence never mentions a car. (The accident could have been a motorcycle accident, or a bicycle accident, or a van hitting a pedestrian, or even a small plane crash.)
Version B (your suggested version) is not incorrect, but it assumes the speaker already knows the accident was an automobile accident involving a single automobile. (This is not outside the realm of possibility; I can imagine, say, two police officers who were already made aware of the accident before they drove past the site.)
Version C (my new version) might be said if two people were standing on the sidewalk viewing the accident site. If there was only one tree nearby and they were both looking at it, the definite article could be used in front of tree, but we might be inclined to use an indefinite article in front of car if the car itself had already been towed away.
Version D (another new version) might be said on the scene, particularly if the car is still smashed against the tree.
If the first sentence was rephrased, though, you'd have a stronger argument for using the car instead of a car:
E: There was a car accident here yesterday. The car hit a tree and the driver was killed.
This would work especially well if there were several trees in the area and the speaker did not know which tree was hit.