Is vegetarian diet better when preparing for half marathon?

It is not better: A good vegi diet is better then a bad meat diet and a good meat diet is better then a bad vegi diet. You have to work harder with a vegi diet to get balenced protein and some vitamins.

No: A distance runner needs about 1.5 grams of complete protein per kilo of ideal body weight, so about 90 grams a day for an average woman. This is difficult to do in a vegan diet, but quite easy if you add fish , shellfish, eggs and/ or milk products. Eat whole, natural unprocessed foods and don't restrict your salt while training outdoors in the summer.

Not better: Carbo-loading is a typical routine just prior to a marathon. Often this consists of a purely vegetarian meal consisting of pasta. During the training phase, a well-balanced vegetarian diet can designed but may not be considered "better" because of the difficulty in ensuring adequate protein, B12, and omega-3-fatty acid intake. Check with a qualified nutritionist before "running" down that road.