In Italian you don't write "di i" or "di le" ("preposizione semplice" followed by a definite article), instead the two combine and form what is called a "preposizione articolata". Here's a scheme:

             il          lo          l'          la          i          gli          le

di        del      dello    dell'    della     dei      degli     delle

a          al         allo      all'       alla       ai       agli       alle

da       dal     dallo    dall'    dalla      dai     dagli     dalle

in        nel      nello    nell'    nella      nei     negli     nelle

su       sul      sullo    sull'    sulla      sui     sugli     sulle

Ciao Saulo,

Guyomar answered your question. Those are called "articoli partitivi" (partitive articles) and the one that is required depends on gender and number of the noun. 

Example: 

word: bambino (child) 

word + definite article: il bambino (the child) - I bambini (the children)

word + articolo partitivo: de + i bambini ---> DEI bambini (some children) 

but if you have 

Example:

word: albero (tree)

word + definite article: l'albero (the tree) - gli alberi (the trees)

word + articolo partitivo: de + gli ---> DEGLI alberi (some trees)

In conclusion, you can see by yourself why the word "bambini" needs a different articolo partitivo compared to the word "alberi".

DEI bambini versus DEGLI alberi 

The same process can be done with DELLE and feminine nouns.

Now, I will tell you something more that you don't find always on Italian grammar books: be careful with uncountable nouns, because if you try to say "some sugar" for example, you need to use the singular masculine partitive article DELLO instead of DEGLI. If are using DEGLI you are basically saying "many types of sugar". That is different. 

If you want to improve your Italian and practice this topic contact me, I am a teacher here on Italki. 

Buona giornata! 

Andrea