Why did the early versions of the Me-109 use a braced stabilizer?
To understand that particular design decision, some background may be necessary.

In the 1920s, Willy Messerschmitt had started a small aircraft factory in Bamberg with limited economical success. When he requested financial help from the Bavarian state, he was forced to merge with another aircraft factory in Augsburg, into Bayerische Flugzeugwerke. Since few of his designs enjoyed commercial success, he was forced to accept the production of other designs under license to keep the workforce of the combined companies occupied.

The only Messerschmitt design which had been accepted by Lufthansa was the M.20, and a close friend of a Lufthansa director was killed in a test flight of the M.20. When that director was promoted to the German air ministry and became responsible for selecting aircraft for building up the new Luftwaffe, he made it clear to Messerschmitt that he would avoid purchasing his designs because he disapproved of his reaction to that fatal accident of the M.20.



Messerschmitt M.20 (picture source)

The most successful type of Messerschmitt's designs was the M.23, a two seater sports plane. It had suffered two accidents in which the tailplane broke off in flight, so struts for the tailplane were added while initially produced aircraft had an unbraced tail. 



Early Messerschmitt M.23a with ABC Scorpion engine (picture source)

Messerschmitt had gained a reputation for aerodynamically clean and very lightweight designs, and accidents of the M.20 and M.23 in which the tail section failed made pilots wary of the structural soundness of his designs. Nevertheless, his aircraft achieved surprising performance with very limited engine power, so he was unofficially asked to contribute a design when a competition for a new standard fighter was announced in 1934. As an outsider, he had to make sure that his design would be accepted by his critics, so he played it safe and braced the tail, as he had done on the M.37 (which was to become the Bf-108 and was in many ways the dress rehearsal to the Bf-109).

That is the real reason for the struts: Messerschmitt had to play it safe to not threaten his chances as an outsider for a very lucrative contract.